In most states, congressional distracts are redrawn after each census by the state legislature. This means that every 10 years the party in control of the legislature is in charge of setting the districts until the next census.
Some states have different ways to draw congressional district boundaries. The most common alternate line drawing plan includes putting bipartisan or non-partisan committees in charge of the redistricting. Usually, the state governor or majority party leaders still control who is appointed to the committee. Sometimes local city council members or leaders are included in the process. Some states are trying to move forward to a system where the citizens themselves are included in the redistricting process instead.